Simultaneous local exact controllability of 1D bilinear Schrödinger equations

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Ann. I. H. Poincaré – AN 31 (2014) 501–529

www.elsevier.com/locate/anihpc

Simultaneous local exact controllability of 1D bilinear Schrödinger equations

Morgan Morancey

^a,b,^∗^,1

aCMLS UMR 7640, Ecole Polytechnique, 91128 Palaiseau, France

bCMLA UMR 8536, ENS Cachan, 61 avenue du Président Wilson, 94235 Cachan, France Received 13 November 2012; received in revised form 25 April 2013; accepted 21 May 2013

Available online 31 May 2013

Abstract

We consider N independent quantum particles, in an infinite square potential well coupled to an external laser field. These particles are modelled by a system of linear Schrödinger equations on a bounded interval. This is a bilinear control system in which the state is theN-tuple of wave functions. The control is the real amplitude of the laser field. ForN=1, Beauchard and Laurent proved local exact controllability around the ground state in arbitrary time. We prove, under an extra generic assumption, that their result does not hold in small time ifN2. Still, forN=2, we prove that local controllability holds either in arbitrary time up to a global phase or exactly up to a global delay. This is proved using Coron’s return method. We also prove that forN3, local controllability does not hold in small time even up to a global phase. Finally, forN=3, we prove that local controllability holds up to a global phase and a global delay.

Keywords:Bilinear control; Schrödinger equation; Simultaneous control; Return method; Non-controllability

1. Introduction 1.1. Main results

We consider a quantum particle in a one dimensional infinite square potential well coupled to an external laser field. The evolution of the wave functionψis given by the following Schrödinger equation

i∂_tψ= −∂_xx² ψ−u(t )μ(x)ψ, (t, x)∈(0, T )×(0,1),

ψ (t,0)=ψ (t,1)=0, t∈(0, T ), (1.1)

whereμ∈H³((0,1),R)is the dipolar moment andu:t∈(0, T )→Ris the amplitude of the laser field. This is a bilinear control system in which the stateψlives on a sphere ofL²((0,1),C). Similar systems have been studied by various authors (see e.g.[6,15,38,44]).

* Correspondence to: CMLS UMR 7640, Ecole Polytechnique, 91128 Palaiseau, France.

E-mail address:Morgan.Morancey@cmla.ens-cachan.fr.

1 The author was partially supported by the “Agence Nationale de la Recherche” (ANR), Projet Blanc EMAQS number ANR-2011-BS01-017-01.

http://dx.doi.org/10.1016/j.anihpc.2013.05.001

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We are interested in simultaneous controllability of system(1.1)and thus we consider, forN∈N^∗, the system i∂_tψ^j= −∂_xx² ψ^j−u(t )μ(x)ψ^j, (t, x)∈(0, T )×(0,1), j∈ {1, . . . , N},

ψ^j(t,0)=ψ^j(t,1)=0, t∈(0, T ), j∈ {1, . . . , N}. (1.2) It is a simplified model for the evolution ofN identical and independent particles submitted to a single external laser field where entanglement has been neglected. This can be seen as a first step towards more sophisticated models.

Before going into details, let us set some notations. In this article, ·,· denotes the usual scalar product on L²((0,1),C)i.e.

f, g = 1 0

f (x)g(x)dx

andSdenotes the unit sphere ofL²((0,1),C). We consider the operatorAdefined by D(A):=H²∩H₀¹

(0,1),C

, Aϕ:= −∂_xx² ϕ.

Its eigenvalues and eigenvectors are λk:=(kπ )², ϕk(x):=√

2 sin(kπ x), ∀k∈N^∗.

The family(ϕk)k∈N^∗is a Hilbert basis ofL²((0,1),C). The eigenstates are defined by Φ_k(t, x):=ϕ_k(x)e⁻^iλ^k^t, (t, x)∈R⁺×(0,1), k∈N^∗.

AnyN-tuple of eigenstates is solution of system(1.2)with controlu≡0. Finally, we define the spaces H₍₀₎^s

(0,1),C :=D

A^s/2

, ∀s >0, endowed with the norm

· H₍₀₎^s :=

_+∞

k=1

k^s·, ϕk² ^1/2

and h^s

N^∗,C :=

a=(ak)_k_∈N∗∈C^N^∗; +∞

k=1

k^sak²<+∞

endowed with the norm ah^s:=

_+∞

k=1

k^sa_k² ^1/2.

Our goal is to control simultaneously the particles modelled by(1.2)with initial conditions

ψ^j(0, x)=ϕ_j(x), x∈(0,1), j∈ {1, . . . , N}, (1.3)

locally around(Φ₁, . . . , Φ_N)using a single control.

Remark 1.1.Before getting to controllability results, it has to be noticed that for any controlv∈L²((0, T ),R), the associated solution of(1.2)satisfies

ψ^j(t), ψ^k(t)

=

ψ^j(0), ψ^k(0)

, ∀t∈ [0, T].

This invariant has to be taken into account since it imposes compatibility conditions between targets and initial conditions.

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The caseN =1 of a single equation was studied, in this setting, in[6, Theorem 1]by Beauchard and Laurent.

They proved exact controllability, inH₍₀₎³ , in arbitrary time, locally aroundΦ₁. Their proof relies on the linear test, the inverse mapping theorem and a regularizing effect. We prove that this result cannot be extended to the caseN=2.

In the spirit of[6], we assume the following hypothesis.

Hypothesis 1.1.The dipolar momentμ∈H³((0,1),R)is such that there existsc >0 satisfying μϕ_j, ϕ_k c

k³, ∀k∈N^∗, ∀j ∈ {1, . . . , N}.

Remark 1.2. In the same way as in [6, Proposition 16], one may prove that Hypothesis 1.1holds generically in H³((0,1),R).

Using[6, Theorem 1],Hypothesis 1.1implies that thejth equation of system(1.2)is locally controllable inH₍₀₎³ aroundΦ_j.

Hypothesis 1.2.The dipolar momentμ∈H³((0,1),R)is such that A:= μϕ₁, ϕ₁

μ2

ϕ₂, ϕ₂

− μϕ₂, ϕ₂ μ2

ϕ₁, ϕ₁

=0.

Remark 1.3.For example,μ(x):=x³satisfies bothHypothesis 1.1 and 1.2. Unfortunately, the caseμ(x):=xstudied in[44]does not satisfy these hypotheses. But, as in[6, Proposition 16], one may prove that Hypotheses1.1 and 1.2 hold simultaneously generically inH³((0,1),R).

Remark 1.4.Hypothesis 1.2implies that there existsj∈ {1,2}such thatμϕ_j, ϕ_j =0. Without loss of generality, whenHypothesis 1.2is assumed to hold, one should consider thatμϕ₁, ϕ₁ =0.

Theorem 1.1.Let N=2 and μ∈H³((0,1),R)be such that Hypothesis1.2hold. Let α∈ {−1,1}be defined by α:=sign(Aμϕ₁, ϕ₁). There existsT_∗>0and ε >0 such that for any T < T_∗, for everyu∈L²((0, T ),R)with u_L²(0,T )< ε, the solution of system(1.2)–(1.3)satisfies

ψ¹(T ), ψ²(T )

=

Φ₁(T ),

1−δ²+iαδ Φ₂(T )

, ∀δ >0.

Thus, underHypothesis 1.2, simultaneous controllability does not hold for(ψ¹, ψ²) around(Φ₁, Φ₂)in small time with small controls. The smallness assumption on the control is inL² norm. This prevents from extending[6, Theorem 1] to the caseN 2. Notice that the proposed target that cannot be reached satisfies the compatibility conditions ofRemark 1.1.

However, when modelling a quantum particle, the global phase is physically meaningless. Thus for anyθ∈Rand ψ¹, ψ²∈L²((0,1),C), the statese^iθ(ψ¹, ψ²)and(ψ¹, ψ²)are physically equivalent. Working up to a global phase, we prove the following theorem.

Theorem 1.2.LetN=2. LetT >0. Letμ∈H³((0,1),R)satisfyHypothesis1.1andμϕ1, ϕ1 = μϕ2, ϕ2. There existsθ∈R,ε0>0and aC¹map

Γ :Oε₀→L²

(0, T ),R where

Oε0:=

ψ_f¹, ψ_f²

∈H₍₀₎³

(0,1),C2

; ψ_f^j, ψ_f^k

=δ_j₌_kand 2 j=1

ψ_f^j−e^iθΦ_j(T )

H₍₀₎³ < ε₀

,

such that for any(ψ_f¹, ψ_f²)∈Oε0, the solution of system(1.2)with initial condition(1.3)and controlu=Γ (ψ_f¹, ψ_f²) satisfies

ψ¹(T ), ψ²(T )

=

ψ_f¹, ψ_f² .

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Remark 1.5.Notice that, usingRemark 1.1, the conditionψ_f^j, ψ_f^k =δ_j₌_kis not restrictive. Indeed, asψ^j(0)=ϕ_j, we can only reach targets satisfying such an orthonormality condition.

Remark 1.6.The same theorem holds with initial conditions(ψ₀¹, ψ₀²)close enough to(ϕ1, ϕ2)inH₍₀₎³ satisfying the constraintsψ₀¹, ψ₀² = ψ_f¹, ψ_f²(seeRemark 4.1in Section4.2).

Working in time large enough we can drop the global phase and prove the following theorem.

Theorem 1.3.LetN=2. Letμ∈H³((0,1),R)satisfyHypothesis1.1and4μϕ1, ϕ1 − μϕ2, ϕ2 =0. There exists T^∗>0such that, for anyT 0, there existsε0>0and aC¹map

Γ :Oε₀,T →L²

0, T^∗+T ,R where

Oε0,T :=

ψ_f¹, ψ_f²

∈H₍₀₎³

(0,1),C2

; ψ_f^j, ψ_f^k

=δ_j₌_kand 2 j=1

ψ^j

f−Φ_j(T )

H₍₀₎³ < ε₀

,

such that for any (ψ_f¹, ψ_f²)∈Oε₀,T, the solution of system (1.2) with initial condition (1.3) and control u= Γ (ψ_f¹, ψ_f²)satisfies

ψ¹

T^∗+T , ψ²

T^∗+T

=

ψ_f¹, ψ_f² . Remark 1.7.Remark 1.6is still valid in this case.

We now turn to the caseN=3. We prove that under an extra generic assumption,Theorem 1.2cannot be extended to three particles. Assume the following hypothesis.

Hypothesis 1.3.The dipolar momentμ∈H³((0,1),R)is such that B: =

μϕ₃, ϕ₃ − μϕ₂, ϕ₂ μ2

ϕ₁, ϕ₁ +

μϕ₁, ϕ₁ − μϕ₃, ϕ₃ μ2

ϕ₂, ϕ₂ +

μϕ₂, ϕ₂ − μϕ₁, ϕ₁ μ2

ϕ₃, ϕ₃

=0.

Remark 1.8.Hypothesis 1.3implies that there existj, k∈ {1,2,3}such thatμϕ_j, ϕ_j = μϕ_k, ϕ_k. Without loss of generality, whenHypothesis 1.3is assumed to hold, one should consider thatμϕ₁, ϕ₁ = μϕ₂, ϕ₂.

Remark 1.9.Again, Hypotheses1.1 and 1.3hold simultaneously generically inH³((0,1),R).

We prove the following theorem.

Theorem 1.4.LetN=3andμ∈H³((0,1),R)be such thatHypothesis1.3hold. Letβ∈ {−1,1}be defined byβ:=

sign(B(μϕ₂, ϕ₂ − μϕ₁, ϕ₁)). There existsT_∗>0andε >0such that, for anyT < T_∗, for everyu∈L²((0, T ),R) withu_L²_{(0,T )}< ε, the solution of system(1.2)–(1.3)satisfies for everyδ >0andν∈R,

ψ¹(T ), ψ²(T ), ψ³(T )

=e^iν

Φ1(T ), Φ2(T ),

1−δ²+iβδ Φ3(T )

.

Thus, in small time, local exact controllability with small controls does not hold forN3, even up to a global phase. The next statement ensures that it holds up to a global phase and a global delay.

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Theorem 1.5.LetN=3. Letμ∈H³((0,1),R)satisfyHypothesis1.1and5μϕ₁, ϕ₁−8μϕ₂, ϕ₂+3μϕ₃, ϕ₃ =0.

There existsθ∈R,T^∗>0such that, for anyT 0, there existsε₀>0and aC¹map Γ :Oε₀,T →L²

0, T^∗+T ,R where

Oε₀,T :=

ψ_f¹, ψ_f², ψ_f³

∈H₍₀₎³

(0,1),C3

; ψ_f^j, ψ_f^k

=δj=kand 3 j=1

ψ^j

f−e^iθΦj(T )

H₍₀₎³ < ε0

,

such that for any (ψ_f¹, ψ_f², ψ_f³)∈Oε₀,T, the solution of system (1.2)with initial condition(1.3)and control u= Γ (ψ_f¹, ψ_f², ψ_f³)satisfies

ψ¹

T^∗+T , ψ²

T^∗+T , ψ³

T^∗+T

=

ψ_f¹, ψ_f², ψ_f³ . Remark 1.10.Remark 1.6is still valid in this case.

1.2. Heuristic

Contrarily to the case N =1, the linearized system around anN-tuple of eigenstates is not controllable when N2. Let us consider, forN=2, the linearization of system(1.2)around(Φ₁, Φ₂),

⎧⎨

⎩

i∂_tΨ^j= −∂_xx² Ψ^j−v(t )μ(x)Φ_j, (t, x)∈(0, T )×(0,1), j∈ {1,2}, Ψ^j(t,0)=Ψ^j(t,1)=0, t∈(0, T ),

Ψ^j(0, x)=0, x∈(0,1).

(1.4) Forj =1,2, straightforward computations lead to

Ψ^j(T )=i

+∞

k=1

μϕ_j, ϕ_k T 0

v(t )e^i(λ^k⁻^λ^j^)tdt Φk(T ). (1.5)

Thus, thanks to Hypothesis 1.1, we could, by solving a suitable moment problem, control any directionΨ^j(T ), Φ_k(T ), fork2 (with a slight abuse of notation for the directionΦ_kof thejth equation). Straightforward computations using(1.5)lead to

Ψ¹(T ), Φ2(T ) +

Ψ²(T ), Φ1(T )

=0.

This comes from the linearization of the invariant (seeRemark 1.1) ψ¹(t), ψ²(t)

= ψ₀¹, ψ₀²

, ∀t∈(0, T ),

and can be overcome (see Section4.2). However,(1.5)also implies that μϕ2, ϕ2

Ψ¹(T ), Φ1(T )

= μϕ1, ϕ1

Ψ²(T ), Φ2(T ) ,

for anyv∈L²((0, T ),R). This is a strong obstacle to controllability and leads toTheorem 1.1(see Section6).

In this situation, where a direction is lost at the first order, one can try to recover it at the second order. This strategy was used for example by Cerpa and Crépeau in[13]on a Korteweg De Vries equation and adapted on the considered bilinear Schrödinger equation(1.1)by Beauchard and the author in[8]. Let, forj∈ {1,2},

⎧⎪

⎨

⎪⎩

i∂_tξ^j= −∂_xx² ξ^j−v(t )μ(x)Ψ^j−w(t)μ(x)Φ_j, (t, x)∈(0, T )×(0,1), ξ^j(t,0)=ξ^j(t,1)=0, t∈(0, T ),

ξ^j(0, x)=0, x∈(0,1).

The main idea of this strategy is to exploit a rotation phenomenon when the control is turned off. However, as proved in[8, Lemma 4], there is no rotation phenomenon on the diagonal directionsξ^j(T ), Φ_j(T )and this power series expansion strategy cannot be applied to this situation.

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Thus, the local exact controllability results in this article are proved using Coron’s return method. This strategy, detailed in[23, Chapter 6], relies on finding a reference trajectory of the nonlinear control system with suitable origin and final positions such that the linearized system around this reference trajectory is controllable. Then, the inverse mapping theorem allows to prove local exact controllability.

As the Schrödinger equation is not time reversible, the design of the reference trajectory(ψ_ref¹ , . . . , ψ_ref^N, u_ref)is not straightforward. The reference controlu_ref is designed in two steps. The first step is to impose restrictive conditions onu_ref on an arbitrary time interval(0, ε)in order to ensure the controllability of the linearized system. Then,u_ref is designed on(ε, T^∗)such that the reference trajectory at the final time coincides with the target. For example, to prove Theorem 1.5, the reference trajectory is designed such that

ψ_ref¹ T^∗

, ψ_ref² T^∗

, ψ_ref³ T^∗

=e^iθ(ϕ₁, ϕ₂, ϕ₃). (1.6)

1.3. Structure of the article

This article is organized as follows. We recall, in Section2, well posedness results.

To emphasize the ideas developed in this article, we start by proving Theorem 1.5. Section3 is devoted to the construction of the reference trajectory. In Section4.1, we prove the controllability of the linearized system around the reference trajectory. In Section4.2, we conclude the return method thanks to an inverse mapping argument.

In Section 5, we adapt the construction of the reference trajectory for two equations leading to Theorems 1.2 and 1.3.

Finally, Section6is devoted to non-controllability results and the proofs ofTheorems 1.1 and 1.4.

1.4. A review of previous results

Let us recall some previous results about the controllability of Schrödinger equations. In[3], Ball, Marsden and Slemrod proved a negative result for infinite dimensional bilinear control systems. The adaptation of this result to Schrödinger equations, by Turinici [45], proves that the reachable set with L² controls has an empty interior in S∩H₍₀₎² ((0,1),C). Although this is a negative result it does not prevent controllability in more regular spaces.

Actually, in[4], Beauchard proved local exact controllability inH⁷using Nash–Moser theorem for a one dimensional model. The proof of this result was simplified, by Beauchard and Laurent in[6], by exhibiting a regularizing effect allowing to apply the classical inverse mapping theorem. In[5], Beauchard and Coron also proved exact controllability between eigenstates for a particle in a moving potential well.

Using stabilization techniques and Lyapunov functions, Nersesyan proved in[42]that Beauchard and Laurent’s result holds globally inH³⁺^ε. Other stabilization results on similar models were obtained in[7,9,38,40,41]by Mir- rahimi, Beauchard, Nersesyan and the author.

Unlike exact controllability, approximate controllability results have been obtained for Schrödinger equations on multidimensional domains. In[14], Chambrion, Mason, Sigalotti and Boscain proved approximate controllability in L², thanks to geometric technics on the Galerkin approximation both for the wave function and density matrices. These results were extended to stronger norms in[12]by Boussaid, Caponigro and Chambrion. Approximate controllability in more regular spaces (containingH³) were obtained by Nersesyan and Nersisyan[43]using exact controllability in infinite time. Approximate controllability has also been obtained by Ervedoza and Puel in[26]on a model of trapped ions.

Simultaneous exact controllability of quantum particles has been obtained on a finite dimensional model in[46]

by Turinici and Rabitz. Their model uses specific orientation of the molecules and their proof relies on iterated Lie brackets. In addition to the results of[14], simultaneous approximate controllability was also studied in [15] by Chambrion and Sigalotti. They used controllability of the Galerkin approximations for a model of different particles with the same control operator and a model of identical particles with different control operators. These simultaneous approximate controllability results are valid regardless of the number of particles considered.

Finally, let us give some details about the return method. This idea of designing a reference trajectory such that the linearized system is controllable was developed by Coron in[18]for a stabilization problem. It was then successfully used to prove exact controllability for various systems: Euler equations in[19,28,30]by Coron and Glass, Navier–

Stokes equations in[17,20,24,27]by Coron, Fursikov, Imanuvilov, Chapouly and Guerrero, Bürgers equations in[16,

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32,34]by Horsin, Glass, Guerrero and Chapouly and many other models such as[21,25,29,31]. This method was also used for a bilinear Schrödinger equation in[4]by Beauchard.

The question of simultaneous exact controllability for linear PDE is already present in the book[37]by Lions. He considered the case of two wave equations with different boundary controls. This was later extended to other systems by Avdonin, Tucsnak, Moran and Kapitonov in[1,2,35].

To conclude, the question of impossibility of certain motions in small time, at stake in this article, for bilinear Schrödinger equations was studied in[8,22]by Coron, Beauchard and the author.

2. Well posedness

First, we recall the well posedness of the considered Schrödinger equation with a source term which proof is in[6, Proposition 2]. Consider

⎧⎨

⎩

i∂tψ (t, x)= −∂_xx² ψ (t, x)−u(t)μ(x)ψ (t, x)−f (t, x), (t, x)∈(0, T )×(0,1),

ψ (t,0)=ψ (t,1)=0, t∈(0, T ),

ψ (0, x)=ψ0(x), x∈(0,1).

(2.1)

Proposition 2.1.Let μ∈H³((0,1),R),T >0,ψ₀∈H₍₀₎³ (0,1),u∈L²((0, T ),R)andf ∈L²((0, T ), H³∩H₀¹).

There exists a unique weak solution of (2.1), i.e. a function ψ∈C⁰([0, T], H₍₀₎³ ) such that the following equality holds inH₍₀₎³ ((0,1),C)for everyt∈ [0, T],

ψ (t )=e⁻^iAtψ₀+i t

0

e⁻^iA(t⁻^{τ )}

u(τ )μψ (τ )+f (τ ) dτ.

Moreover, for every R >0, there existsC=C(T , μ, R) >0 such that, ifu_L²(0,T )< R, then this weak solution satisfies

ψ_C0([0,T],H₍₀₎³ )C ψ₀_H³

(0)+ f_L2((0,T ),H³∩H₀¹)

. Iff ≡0, then

ψ (t )

L²(0,1)= ψ₀L²(0,1), ∀t∈ [0, T].

3. Construction of the reference trajectory for three equations

The goal of this section is the design of the following family of reference trajectories to proveTheorem 1.5.

Theorem 3.1.LetN=3. Letμ∈H³((0,1),R)satisfyHypothesis1.1and5μϕ₁, ϕ₁−8μϕ₂, ϕ₂+3μϕ₃, ϕ₃ =0.

LetT1>0be arbitrary,ε∈(0, T1)andε1∈(^ε₂, ε). There existη >0,C >0such that for everyη∈(0, η), there exist T^η> T1,θ^η∈Randu^η_ref ∈L²((0, T^η),R)with

u^η_ref

L²(0,T^η)Cη (3.1)

such that the associated solution(ψ_ref^1,η, ψ_ref^2,η, ψ_ref^3,η)of(1.2)–(1.3)satisfies μψ_ref^1,η(ε₁), ψ_ref^1,η(ε₁)

= μϕ₁, ϕ₁ +η, μψ_ref^2,η(ε1), ψ_ref^2,η(ε1)

= μϕ2, ϕ2, μψ_ref^3,η(ε₁), ψ_ref^3,η(ε₁)

= μϕ₃, ϕ₃, (3.2)

μψ_ref^1,η(ε), ψ_ref^1,η(ε)

= μϕ1, ϕ1, μψ_ref^2,η(ε), ψ_ref^2,η(ε)

= μϕ₂, ϕ₂ +η, μψ_ref^3,η(ε), ψ_ref^3,η(ε)

= μϕ₃, ϕ₃, (3.3)

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and

ψ_ref^1,η T^η

, ψ_ref^2,η T^η

, ψ_ref^3,η T^η

=e^iθ^η(ϕ₁, ϕ₂, ϕ₃). (3.4)

Remark 3.1. For any T 0, u^η_ref is extended by zero on (T^η, T^η+T ). Thus, there exists C >0 such that, u^η_refL²(0,T^η+T )Cη,(3.2), (3.3)are satisfied and

ψ_ref^1,η

T^η+T , ψ_ref^2,η

T^η+T , ψ_ref^3,η

T^η+T

=e^iθ^η

Φ₁(T ), Φ₂(T ), Φ₃(T ) .

Remark 3.2.The choice of a parameterηsufficiently small together with conditions(3.2) and (3.3)will be used in Section4.1to prove the controllability of the linearized system around the reference trajectory. The controlu^η_ref will be designed on(0, T1)and extended by zero on(T1, T^η).

The proof ofTheorem 3.1is divided in two steps: the construction ofu^η_ref on(0, ε)to prove(3.2) and (3.3)and then, the construction on(ε, T₁)to prove(3.4). This is what is detailed in the next subsections.

3.1. Construction on(0, ε)

Letu^η_ref ≡0 on[0,₂^ε). We prove the following proposition.

Proposition 3.1.Letμ∈H³((0,1),R)satisfyHypothesis1.1. There existsη^∗>0and aC¹map ˆ

Γ : 0, η^∗

→L² ε

2, ε

,R

,

such thatΓ (0)ˆ =0and for anyη∈(0, η^∗), the solution(ψ_ref^1,η, ψ_ref^2,η, ψ_ref^3,η)of system(1.2)with controlu^η_ref := ˆΓ (η) and initial conditionsψ_ref^j,η(^ε₂)=Φ_j(^ε₂), forj=1,2,3, satisfies(3.2)and(3.3).

Proof of Proposition 3.1. UsingProposition 2.1, it comes that the map Θ˜ : L²

ε 2, ε

,R

→ R³×R³ u → Θ˜1(u),Θ˜2(u) where

Θ˜₁(u):=

μψ^j(ε₁), ψ^j(ε₁)

− μϕ_j, ϕ_j

j=1,2,3, and

Θ˜₂(u):=

μψ^j(ε), ψ^j(ε)

− μϕ_j, ϕ_j

j=1,2,3, is well defined,C¹, satisfiesΘ(0)˜ =0 and

dΘ(0).v˜ =

2

μΨ^j(ε₁), Φ_j(ε₁)

1j3,

2

μΨ^j(ε), Φ_j(ε)

1j3

, (3.5)

where (Ψ¹, Ψ², Ψ³) is the solution of (1.4) on the time interval (^ε₂, ε) with control v and initial conditions Ψ^j(^ε₂,·)=0. Let us prove that dΘ(0)˜ is surjective; then the inverse mapping theorem will give the conclusion.

Letγ =(γ_j)₁_j₆∈R⁶andK4. ByProposition A.1(seeAppendix A), there existv₁∈L²((^ε₂, ε₁),R)and v2∈L²((ε1, ε),R)such that

ε1

ε 2

v₁(t)e^i(λ^k⁻^λ^j^)tdt=0, ∀k∈N^∗\{K}, ∀1j 3,

(9)

ε₁

ε 2

v₁(t)e^i(λ^K⁻^λ^j^)tdt=e^i(λ^K⁻^λ^j^)ε¹γ_j

2iμϕj, ϕK², ∀1j3, ε

ε₁

v₂(t)e^i(λ^k⁻^λ^j^)tdt=0, ∀k∈N^∗\{K}, ∀1j3, ε

ε1

v₂(t)e^i(λ^K⁻^λ^j^)tdt=e^i(λ^K⁻^λ^j^)εγ3+j

2iμϕ_j, ϕ_K² − e^i(λ^K⁻^λ^j^)ε¹γj

2iμϕ_j, ϕ_K², ∀1j3.

Notice that the moments associated to redundant frequencies in the previous moment problem are all set to the same value and, asK4, the frequenciesλ_K−λ_jfor 1j3 are distinct. Letv∈L²(^ε₂, ε)be defined byv₁on(^ε₂, ε₁) and byv2on(ε1, ε). Straightforward computations lead to dΘ(0).v˜ =γ. 2

3.2. Construction on(ε, T₁)

For anyj ∈N^∗, letPj be the orthogonal projection ofL²((0,1),C)onto Span_C(ϕ_k, kj+1)i.e.

Pj(ψ ):= ^+∞

k=j+1

ψ, ϕ_kϕ_k.

The goal of this subsection is the proof of the following proposition.

Proposition 3.2. Let 0< T₀< T_f. Letμ∈H³((0,1),R) satisfy Hypothesis 1.1 and 5μϕ₁, ϕ₁ −8μϕ₂, ϕ₂ + 3μϕ₃, ϕ₃ =0. There existδ >0and aC¹map

Γ˜T₀,T_f : ˜Oδ,T₀→L²

(T0, Tf),R with

O˜δ,T₀:=

ψ₀¹, ψ₀², ψ₀³

∈

S∩H₍₀₎³ (0,1)3

; 3 j=1

ψ₀^j−Φj(T0)

H₍₀₎³ < δ

,

such thatΓ˜_T₀_,T_f(Φ1(T0), Φ2(T0), Φ3(T0))=0and, if(ψ₀¹, ψ₀², ψ₀³)∈ ˜Oδ,T0, the solution(ψ¹, ψ², ψ³)of system(1.2) with initial conditionsψ^j(T0,·)=ψ₀^j, forj=1,2,3, and controlu:= ˜ΓT₀,T_f(ψ₀¹, ψ₀², ψ₀³)satisfies

P1

ψ¹(T_f)

=P2

ψ²(T_f)

=P3

ψ³(T_f)

=0, (3.6)

ψ¹(T_f), Φ₁(T_f)5

ψ²(T_f), Φ₂(T_f)8

ψ³(T_f), Φ₃(T_f)3

=0. (3.7)

Remark 3.3.The conditions(3.6) and (3.7)will be used in the next subsection to prove(3.4). Eq.(3.7)will be used to define the global phaseθ^η.

Proof of Proposition 3.2. Let us define the following space X₁:=

(φ₁, φ₂, φ₃)∈H₍₀₎³

(0,1),C3

; φ_j, ϕ_k =0, for 1kj3 . We consider the following end-point map

Θ_T₀_,T_f :L²((T₀, T_f),R)×H₍₀₎³ (0,1)³→H₍₀₎³ (0,1)³×X₁×R, defined by

ΘT₀,T_f

u, ψ₀¹, ψ₀², ψ₀³ :=

ψ₀¹, ψ₀², ψ₀³,P1

ψ¹(Tf) ,P2

ψ²(Tf) ,P3

ψ³(Tf) ,

ψ¹(T_f), Φ₁(T_f)5

ψ²(T_f), Φ₂(T_f)8

ψ³(T_f), Φ₃(T_f)3

(10)

where(ψ¹, ψ², ψ³)is the solution of(1.2)with initial conditionψ^j(T₀,·)=ψ₀^jand controlu. Thus, we have Θ_T₀_,T_f

0, Φ1(T0), Φ2(T0), Φ3(T0)

=

Φ1(T0), Φ2(T0), Φ3(T0),0,0,0,0 .

Proposition 3.2is proved by application of the inverse mapping theorem toΘ_T₀_,T_f at the point(0, Φ1(T₀), Φ₂(T₀), Φ₃(T₀)).

Using the same arguments as in[6, Proposition 3], it comes thatΘ_T₀_,T_f is aC¹map and that dΘT₀,T_f

0, Φ1(T0), Φ2(T0), Φ3(T0) .

v, Ψ₀¹, Ψ₀², Ψ₀³

=

Ψ₀¹, Ψ₀², Ψ₀³,P1

Ψ¹(T_f) ,P2

Ψ²(T_f) ,P3

Ψ³(T_f) , 5

Ψ¹(T_f), Φ₁(T_f)

−8

Ψ²(T_f), Φ₂(T_f) +3

Ψ³(T_f), Φ₃(T_f) ,

where (Ψ¹, Ψ², Ψ³) is the solution of (1.4) on the time interval (T₀, T_f) with control v and initial conditions Ψ^j(T₀,·)=Ψ₀^j.

It remains to prove that dΘT0,Tf(0, Φ1(T₀), Φ₂(T₀), Φ₃(T₀)):L²((T₀, T_f),R)×H₍₀₎³ (0,1)³→H₍₀₎³ (0,1)³× X₁×Radmits a continuous right inverse.

Let(Ψ₀¹, Ψ₀², Ψ₀³)∈H₍₀₎³ (0,1)³,(ψ_f¹, ψ_f², ψ_f³)∈X₁andr∈R. Straightforward computations lead to

Ψ^j(T_f)=^+∞

k=1

Ψ₀^j, Φ_k(T0)

+iμϕ_j, ϕ_k

Tf

T₀

v(t )e^i(λ^k⁻^λ^j^)tdt Φ_k(T_f).

Findingv∈L²((T₀, T_f),R)such that Pj

Ψ^j(T_f)

=ψ_f^j, ∀j∈ {1,2,3},

5

Ψ¹(T_f), Φ₁(T_f)

−8

Ψ²(T_f), Φ₂(T_f) +3

Ψ³(T_f), Φ₃(T_f)

=r, is equivalent to solving the following trigonometric moment,∀j=1,2,3,∀kj+1,

Tf

T₀

v(t )e^i(λ^k⁻^λ^j^)tdt= 1 iμϕ_j, ϕ_k

ψ_f^j, Φ_k(T_f)

−

Ψ₀^j, Φ_k(T0) ,

T_f

T₀

v(t )dt=r− (5Ψ₀¹, Φ₁(T₀) −8Ψ₀², Φ₂(T₀) +3Ψ₀³, Φ₃(T₀))

5μϕ₁, ϕ₁ −8μϕ₂, ϕ₂ +3μϕ₃, ϕ₃ . (3.8)

UsingProposition A.1and the hypotheses onμ, this ends the proof ofProposition 3.2. 2 3.3. Proof ofTheorem 3.1

Letδ >0 be the radius defined inProposition 3.2 withT₀=εandT_f =T₁. Forη >0 we define the following control

u^η_ref(t):=

⎧⎪

⎨

⎪⎩

0 fort∈(0,^ε₂),

ˆ

Γ (η) fort∈(^ε₂, ε),

Γ˜ε,T₁(ψ_ref^1,η(ε), ψ_ref^2,η(ε), ψ_ref^3,η(ε)) fort∈(ε, T1),

(3.9)

whereΓˆ andΓ˜ are defined respectively inProposition 3.1 and 3.2. We prove that, forηsmall enough, this control satisfies the conditions ofTheorem 3.1.

Proof of Theorem 3.1. The proof is decomposed into two parts. First, we prove that there exists η >0 such that forη∈(0, η),u^η_ref is well defined, satisfiesu^η_refL²(0,T1)Cηand the conditions(3.2), (3.3)are satisfied. Then, we prove the existence ofT^η>0 andθ^η∈Rsuch that ifu^η_ref is extended by 0 on(T₁, T^η), the condition(3.4)is satisfied.